Question: Solve for $x$ and $y$ using elimination. $\begin{align*}-2x-6y &= 9 \\ -x+2y &= 1\end{align*}$
Solution: We can eliminate $x$ when its corresponding coefficients are negative inverses. Recalling our knowledge of least common multiples, multiply the top equation by $-1$ and the bottom equation by $2$ $\begin{align*}2x+6y &= -9\\ -2x+4y &= 2\end{align*}$ Add the top and bottom equations. $10y = -7$ Divide both sides by $10$ and reduce as necessary. $y = -\dfrac{7}{10}$ Substitute $-\dfrac{7}{10}$ for $y$ in the top equation. $-2x-6( -\dfrac{7}{10}) = 9$ $-2x+\dfrac{21}{5} = 9$ $-2x = \dfrac{24}{5}$ $x = -\dfrac{12}{5}$ The solution is $\enspace x = -\dfrac{12}{5}, \enspace y = -\dfrac{7}{10}$.